Page - (000023) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

2.3 ExponentialDecayandOscillations 13 2.3 ExponentialDecayandOscillations Twosimpleexamplesillustrate thematchbetweenstandardmodelsofdynamicsand the transfer functionexpressions.First, the simplestfirst-orderdifferential equation in x(t) forcedby the inputu(t),with initial condition x(0)=0, isgivenby x˙+ax =u, (2.7) whichhas the solution x(t)= ∫ t 0 e−aτu(t−τ)dτ. (2.8) This process describes how x accumulates over time, as inputs arrive at each time pointwith intensityu, and x decaysat ratea. If the input into this system is the impulseorDiracdelta function,u(t)dt =1at t =0andu(t)=0 forall other times, then x(t)= e−at. If the input is theunit step function,u(t)=1for t ≥0andu(t)=0for t <0, then x(t)= 1 a ( 1−e−at) . Many processes follow the basic exponential decay in Eq.2.8. For example, a quantityuofamoleculemayarrive inacompartmentateachpoint in timeandthen decayatrateawithinthecompartment.Atanytime,thetotalamountofthemolecule in the compartment is the sumof the amounts that arrived at each time in the past, u(t−τ),weightedby the fraction that remainsafterdecay,e−aτ. Theprocess inEq.2.7correspondsexactly to the transfer function P(s)= 1 s+a, (2.9) inwhich theoutput is equivalent to the internal state, y≡ x. Inthesecondexample,anintrinsicprocessmayoscillateataparticularfrequency, ω0, describedby x¨+ω20x =u. This systemproduces output x = sin(ω0t) foru=0andan initial condition along the sinecurve.Thecorresponding transfer function is P(s)= ω0 s2+ω20 .